Coherent states, similar to the canonical coherent states of the harmonic oscillator, labeled by quaternions are established on the right and left quaternionic Hilbert spaces. On the left quaternionic Hilbert space reproducing kernels are established. As was claimed by Adler and Millard (J. Math. Phys. 1997 38 2117-26) it is proved that these coherent states cannot be realized as a group action on a quaternionic Hilbert space. As an extension of the complex Hermite polynomials, quaternionic Hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions.

The energy levels, generally known as the Landau levels, which characterize the motion of an electron in a constant magnetic field, are those of the onedimensional harmonic oscillator, with each level being infinitely degenerate. We show in this paper how the associated von Neumann algebra of observables displays a modular structure in the sense of the Tomita-Takesaki theory, with the algebra and its commutant referring to the two orientations of the magnetic field. A Kubo-Martin-Schwinger state can be built which, in fact, is the Gibbs state for an ensemble of harmonic oscillators. Mathematically, the modular structure is shown to arise as the natural modular structure associated with the Hilbert space of all Hilbert-Schmidt operators.

A class of vector coherent states is derived with multiple of matrices as vectors in a Hilbert space, where the Hilbert space is taken to be the tensor product of several other Hilbert spaces. As examples vector coherent states with multiple of quaternions and octonions are given. The resulting generalized oscillator algebra is briefly discussed. Further, vector coherent states for a tensored Hamiltonian system are obtained by the same method. As particular cases, coherent states are obtained for tensored Jaynes-Cummings type Hamiltonians and for a two-level two-mode generalization of the Jaynes-Cummings model.

Abstract. There is a generalized oscillator algebra associated with every class of orthogonal polynomials {Ψn(x)} ∞ n=0 , on the real line, satisfying a three term recurrence relation xΨn(x) = bnΨ n+1 (x) + b n−1 Ψ n−1 (x), Ψ 0 (x) = 1, b −1 = 0. This note presents necessary and sufficient conditions on bn for such algebras to be of finite dimension. As examples, we discuss the dimensions of oscillator algebras associated with Hermite, Legendre and Gegenbauer polynomials. Some remarks on the dimensions of oscillator algebras associated with multi-boson systems are also presented.

Using the orthonormality of the 2D-Zernike polynomials, reproducing kernels, reproducing kernel Hilbert spaces, and ensuring coherent states attained. With the aid of the so-obtained coherent states, the complex unit disc is quantized. Associated upper symbols, lower symbols and related generalized Berezin transforms also obtained. A number of necessary summation formulas for the 2D-Zernike polynomials proved.